Edge

In graph theory, an edge ^[1][2] is one of the fundamental objects making up a graph, together with the vertices it connects. Each edge joins together a pair of vertices. The edges of a graph $G$ form its edge set $E(G)$.

Definition

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Definition 1 (Edge)

Let $G=(V,E)$ be a graph. An edge is an element $e\in E$ of the edge set that connects a pair of vertices $u,v\in V$ of the vertex set.

The vertices $u$ and $v$ are called the endpoints of the edge. Two vertices joined by an edge are called adjacent, and the edge is said to be incident to each of them.

Variants

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Edges vary based on their structural and numerical properties.

A fundamental distinction depends on whether the connection between vertices carries a direction:

• Undirected edge $\{u,v\}$: connects the vertices $u$ and $v$, called endpoints.
• Directed edge $(u,v)$: connects an ordered pair of vertices. The two endpoints are distinguished by name as the tail $u$ and the head $v$.

Edges further vary in their multiplicity and self-connectivity properties:

• Parallel edge: one of multiple edges that connect the same pair of vertices.
• Loop: an edge that connects a vertex to itself.

Edges may also be assigned additional attributes:

• Weighted edge: an edge with a numerical value (weight) assigned to it.
• Labeled edge: an edge with a label or name assigned to it.

References

1. [1]

   “Glossary of graph theory”, Wikipedia, Available: https://en.wikipedia.org/wiki/Edge_(graph_theory), Accessed: 2026-07-14 ↩

2. [2]

   “Graph Edge”, Wolfram MathWorld, Available: https://mathworld.wolfram.com/GraphEdge.html, Accessed: 2026-07-14 ↩